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The paper establishes conditions under which there are exact linearrepresentations of nonlinear partial differential equations (Cauchy problems).By introducing a certain linear operator $A$, it is shown that under theseconditions there are three equivalent equations (one linear and two nonlinear),while the formal operator solution is common for all of them and is theexpansion of the desired function $v(t,x)$ into a formal Taylor series.  Using the Borel-Whitney lemma, which states that any smooth function $v(t,x)$in a neighborhood of a point is defined by its formal Taylor series, we obtainthe following statement. If all parameters of the operator $A$ are smoothfunctions, then there exists a smooth function $\tilde{v}(t,x)$, which for$t=0$ has the same power expansion as $v(t,x) $ and this function solves allequivalent equations.  The Navier-Stokes and Euler equations are considered as a non-trivialexamples of this approach.

Exact linear representations of the nonlinear Cauchy problems and their smooth solutions